JORDAN'S LEMMA
Jordan’s lemma explains the behaviour of a contour integral on the semicircular upper arc and is frequently used along the residue theorem to evaluate such integrals.
Consider the upper semicircle \(C_R=\{Re^{i\theta}|\theta\in[0,\pi]\}\) and a continuous function \(f:C_R\to\mathbb{C}\). If \(f(z)=e^{i\lambda z}g(z)\) for some function \(g\) and \(\lambda\in\mathbb{R}^+\), then the contour integral is bounded.
\[\displaystyle\left|\int_{C_R}f(z)\ \mathrm{d}z\right|\leq\dfrac{\pi}{\lambda}M_R\ \text{where } M_R:=\max_{\theta\in[0,\pi]}\left|g(Re^{i\theta})\right|\]
#Jordan #JordanLemma #Lemma #Semicircle #ContourIntegral #ResidueTheorem
